Intrinsic spin accumulation in the magnetic spin Hall effect

TL;DR

The paper addresses the ambiguity of spin current definitions in systems with spin–orbit coupling and/or magnetic order by formulating an intrinsic spin-accumulation coefficient (SAC) as the bulk spin response to an electric-field gradient. Using Kubo linear response and a gauge-covariant Bloch-wave formalism, it isolates an intrinsic contribution $g_{sa}^{( ext{II}) ij}$ that can be nonzero in magnetic insulators, in contrast to the magnetic spin Hall conductivity. The authors apply the framework to altermagnets RuO$_{2}$ and MnF$_{2}$, revealing symmetry-determined SAC components and the distinct roles of SOC and insulating behavior. They further relate the intrinsic SAC to the spin magnetic octupole moment and discuss experimental implications for bulk boundary spin accumulation and interface spin-torque phenomena. Overall, the work provides a robust theoretical route to predict and interpret boundary spin accumulation in magnetic materials, particularly in insulating altermagnets.

Abstract

The magnetic spin Hall effect is a time-reversal-odd phenomenon in which spin current is induced by the charge current. In the presence of a spin-orbit coupling and/or noncolinear magnetism, however, spin current is not uniquely defined. Instead, we study an intrinsic response of spin to an electric field gradient that describes the spin accumulation at the boundaries of magnetic systems. We derive a generic formula expressed by Bloch wave functions and apply it to a minimal model for representative altermagnets, RuO$_{2}$ and MnF$_{2}$. Our results show that the intrinsic spin accumulation can be nonzero in magnetic insulators in sharp contrast to the magnetic spin Hall conductivity.

Figures (8)

• Figure 1: Band structure for RuO$_{2}$. Red and blue circles represent spin-up and spin-down dominant bands, respectively. Altermagnetic spin splitting is found in $M$-$\Gamma$ and $A$-$Z$ lines.
• Figure 2: Chemical potential dependence of the intrinsic SAC for RuO$_{2}$ in the absence of the SOCs. Only $g_{sz}^{(\mathrm{II}) xy} = g_{sz}^{(\mathrm{II}) yx}$ is allowed.
• Figure 3: Chemical potential dependence of the intrinsic SAC for RuO$_{2}$. (a) $g_{sx}^{(\mathrm{II}) yz} = g_{sy}^{(\mathrm{II}) xz}$, (b) $g_{sx}^{(\mathrm{II}) zy} = g_{sy}^{(\mathrm{II}) zx}$, and (c) $g_{sz}^{(\mathrm{II}) xy} = g_{sz}^{(\mathrm{II}) yx}$.
• Figure 4: Chemical potential dependence of the conventional SAC for RuO$_{2}$. (a) $\gamma_{sx}^{\space yz} = -\gamma_{sy}^{\space xz}$, (b) $\gamma_{sx}^{\space zy} = -\gamma_{sy}^{\space zx}$, and (c) $\gamma_{sz}^{\space xy} = -\gamma_{sz}^{\space yx}$.
• Figure 5: Band structure for MnF$_{2}$. Red and blue circles represent spin-up and spin-down dominant bands, respectively. Altermagnetic spin splitting is found in $M$-$\Gamma$ and $A$-$Z$ lines.